Journal of Algebraic Combinatorics 5 (1996), 37-46
© 1996 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
Delaunay Transformations of a Delaunay Polytope*
MONIQUE LAURENT
LIENS, Ecole Normale Superieure, 45 rued'Ulm, 75230 Paris Cedex 05, France
Received July 13, 1994; Revised December 21, 1994
Abstract. Let P be a Delaunay polytope in E". Let T[P) denote the set of affine bijections f of R" for which
f(P) is again a Delaunay polytope. The relation: f ~ g if f, g differ by an orthogonal transformation and/or
a translation is an equivalence relation on T(P). We show that the dimension (in the topological sense) of the
quotient set T ( P ) / ~ coincides with another parameter of P, namely, with its rank.
Let V denote the set of vertices of P and let dp denote the distance on V defined by dp(u, v) = \\u - u||2
for u, v 6 V. Assouad [1] shows that dp belongs to the cone H|v| := (d | y].. vfV bubvd(u, v) < 0 for b e
Zv with 5ZueV bu = 1}. Then, the rank of P is defined as the dimension of the smallest face of the cone ~H.\v\
that contains dp.
Keywords:
1.
Delaunay polytope, affine transformation, lattice, dimension, hypermetric
Introduction
This paper is motivated by a question of Billera (private communication, 1994), who asked
whether the notion of rank of a Delaunay polytope P, which is defined in [3] in terms of
a certain cone H | v | , can be expressed in a more intrinsic way as an invariant of a set of
transformations of P. We give a positive answer to this question. Namely, we show that
the rank of P is equal to the dimension (in the topological sense) of the set consisting of the
affine bijections f (up to Euclidian motions) for which f(P) is again a Delaunay polytope.
In this result, we use the notion of dimension of a topological space. This notion was
defined at the beginning of the twentieth century, in particular, after the works of Brouwer,
Monger, Urysohn; see, e.g., [5].
Namely, for a topological space X, its dimension dim(X) is defined in the following
way. If, for any open sets Gi (1 < i < s) such that X = Ui«<* G,, there exist open
sets Hj (1 < j < t) such that X = Ui</</ Hj> each Hj is contained in some Gi, and
the intersection of any n + 2 Hj's is empty, then dim(X) < n. If dim(X) < n but not
dim(X) < n - 1 then dim(X) = n.
This concept generalizes the usual notion of dimension for a Euclidian space or a polyhedron.
We do not need, however, to know the precise definition of this notion of dimension. We will
only use the fact that the dimension is a topological invariant, i.e., that two homeomorphic
topological spaces have the same dimension.
We recall the definitions for a Delaunay polytope and its rank in Sections 1.1 and 1.2.
*AMS Subject Classification (1991): 11H06,52C07.
38
LAURENT
1.1. Delaunay transformations
Let P be an n-dimensional polytope in Rn with set of vertices V. Then, P is said to be a
Delaunay polytope if the following conditions hold:
• The set L := {^,veV bvv | b € Zv and £u,=v bv = 1} is a lattice (i.e., there exists a
nonempty ball centered at each lattice point that contains no other lattice point).
• There exists a sphere S with center c and radius r such that
with equality in (1.1) if and only if x is a vertex of P.
In other words, no lattice point lies in the interior of the ball whose boundary sphere is S
and the lattice points lying on S are precisely the vertices of P. In particular, P is inscribed
on the sphere 5. (Here, \\x\\ = ^/XTX denotes the Euclidian norm of x e R n .)
Delaunay poly topes were introduced by Voronoi [6, 7] (they are also called L-poly topes
in the literature). They are closely related to the well known Voronoi polytopes. Namely,
the vertices of the Voronoi polytope at a lattice point u are precisely the centers of the
Delaunay polytopes that have M as a vertex.
Let f : Rn -+ Rn be an affine bijection. In general, f ( P ) is not a Delaunay polytope.
For instance, an equilateral triangle is a Delaunay polytope while a triangle with a right
angle is not a Delaunay polytope; in fact, a triangle is a Delaunay polytope if and only if
it has no obtuse angle. We call / a Delaunay transformation of P if f ( P ) is a Delaunay
polytope and we let T(P) denote the set of all Delaunay transformations of P. Observe that
all translations, orthogonal transformations, and homotheties are Delaunay transformations
of P. Given two affine bijections f, g of Rn, write
if f, g differ only by an orthogonal transformation or a translation, i.e., if there exist an
orthogonal transformation h of Rn and a e Rn such that g(x) = h(f(x))+a for all x e Rn.
The relation ~ is an equivalence relation on T(P). Let T(P)/ ~ denote the quotient space
of T(P) by ~.
Our goal in this paper is to evaluate the dimension (in the topological sense) of the set
T(P)/~. In fact, the set T(P)/~ can be more simply defined in terms of matrices.
Clearly, we can suppose that the origin is a vertex of P (else, replace P by a translate
of it). Then, every equivalence class of T(P)/ ~ contains a representative /, which maps
the origin on the origin. Hence, f can be identified with the nonsingular matrix A, which
represents f in the canonical basis of Rn. Given two n x n matrices A, B, write
When restricted to the set GL(n) of the n x n nonsingular matrices, the definition of the
relation ~ from (1.3) is coherent with the one given in (1.2). Namely, for A, B e GL(n),
A ~ B if A = UB for some orthogonal matrix U. Set
DELAUNAY TRANSFORMATIONS OF A DELAUNAY POLYTOPE
39
where A (P) := (Ax | x e P} denotes the image of P under A. From the discussion above,
it follows that
As an example, we describe below the Delaunay transformations of the triangle and of
the cube.
Example 1.6 Consider the triangle a2 with vertices V0 = (0, 0), v\ = (2, 0) and v3 =
(1, 2); it is a Delaunay polytope. Every class of the set T(a 2 )/ ~ admits a representative
of the form A — ("
), where a, b, c satisfy:
Indeed, up to rotation, every Delaunay transformation A of a2 can be supposed to leave
the x-axis invariant and, hence, has the form (^ ). The conditions on a, b, c express the
fact that each angle of the triangle A(a 2 ) is acute. Geometrically, this means that the point
A(v2) should lie in the shaded region shown in the figure below.
This shows that there are three degrees of freedom for the parameters of a Delaunay transformation (up to orthogonal transformation) of a2'< in other words, the set T(a2)/ ~ has
dimension 3. More generally, an easy induction shows that, for the n-dimensional simplex
an, T(a n )/~ has dimension ( n+1 ).
Example 1.7 Consider now the square y2 = [0, 1]2. It is easy to see that each class of
T(Y2)/ ~ has a representative of the form (a,0), where a, b > 0. Hence, the set T(y2)/ ~
has dimension 2. More generally, for the n-dimensional cube yn, T(y n )/ ~ has dimension n.
40
LAURENT
1.2. Rank of a Delaunay polytope
Let P be a Delaunay polytope with set of vertices V. We consider the cone H|v| in the
space R ( 2) (indexed by the pairs of elements of V) defined by the inequalities:
for all b e Zv such that £ U6v bu = 1. (We suppose that the elements of V are ordered and
u < v refers to that order.) We use the notation H|v| as the cone H|v| depends only on |V|.
The cone H\v\ is known as the hypermetric cone. Note that H\v\ is defined by infinitely
many inequalities. However, it is shown in [4] that H\v\ is a polyhedral cone, i.e., that a
finite subset of the inequalities (1.8) suffices to describe H\v\.
One can observe that one point dP belonging to the hypermetric cone U\v\ can be constructed from P. Namely, set
for u, v e V. Then, the vector dp := (dp(u, v))u,vev,u<v belongs to H\v\. To see it, take
b 6 Zv such that ^L,U€V bu = 1. Let c and r denote the center and radius of the sphere
circumscribing P. Then,
where the last inequality follows from (1.1). This property was observed by Assouad [1].
Assouad proved, moreover, that, conversely, every point of the cone H\v\ arises in some
sense from a Delaunay polytope. More precisely, let d be an arbitrary point of the cone
H\v\. Then, there exists a Delaunay polytope Q with set of vertices W and a mapping
<f>: V ->• W such that
for all u, v € V. We refer to [2] for a detailed survey on the connections between Delaunay
polytopes and the hypermetric cone.
This leads to the following notion of rank for a Delaunay polytope, introduced in [3].
Definition 1.10 Let P be a Delaunay polytope with set of vertices V and let dp denote
the point of the cone H\v\ defined by (1.9). Then, the rank of P is defined as the dimension
of the smallest face of H\v\ that contains dP.
For instance, the it-simplex ctn has rank ("^
) and the n-cube has dimension n (see [3]).
41
DELAUNAY TRANSFORMATIONS OF A DELAUNAY POLYTOPE
1.3. The main result
The following is the main result of the paper. The proof is given in Section 2.
Theorem 1.11 Let P be a Delaunay polytope. Then, its rank is equal to the dimension
of the quotient set T(P)/~ of Delaunay transformations of P.
Remark 1.12 Note that the dimension of T(P)/ ~ is always greater or equal to 1, as the
homotheties are Delaunay transformations of any Delaunay polytope. It is shown in [3]
that P has rank 1 if and only if the homotheties are the only Delaunay transformations of
P (up to translations and orthogonal transformations), i.e., if the dimension of T(P)/~ is
equal to 1. Hence, Theorem 1.11 holds for rank one Delaunay poly topes. Several examples
of rank one Delaunay poly topes are described in [3].
2. Proof of Theorem 1.11
In what follows, P denotes an n-dimensional Delaunay polytope in Rn with set of vertices
V and admitting the origin as a vertex. As the hypermetric cone H\v\ is a polyhedral cone,
there exists afiniteset BP c {b e Zv | £ U6V bu = 1} such that
{ x elR^' | 2_,b b x
V
V
u v uv
<0
1
for all b e BP 1.
U.UEV
H<u
(2.1)
J
Let dp denote the point of H\v\ defined by (1.9). Let FP denote the smallest face of the
cone H\v\ that contains dp. Then, FP is defined by
for some subset Ap c BP.
2.1. A characterization of the Delaunay transformations of P
We start with an easy result of linear algebra. We use the following notation: For two n x n
matrices A, B,
denotes the usual scalar product, with an n x n matrix being viewed as an n2-vector. Recall
the identity: XT Ax = (A, XXT) for an n x n matrix A and x e Rn.
Lemma 2.3 Let x 1 , . . . , xn be n linearly independent vectors in Rn. Then, the system
S = {x t xT(1 < i < n), (Xj - Xj)(Xj - X J ) T ( 1 < i < j < n)} is linearly independent.
Proof: As 5 consists of n + (n2)) = (" ^ ) elements, it suffices to show that, if X is a
symmetric n x n matrix orthogonal to all members of 5, then X is the zero matrix. By
42
LAURENT
assumption, (X, xfxT) = xJXxt = 0 for i = 1 , . . . , n, and (X, (xi, - xj)(xi - x j ) T ) =
(Xi-XjfX(xi-Xj)
= 0, implying that xfXxj+xJ Xxt = 0 for 1 < i < j < n. We check
that xTXx = 0 for all x e Rn. Indeed, let x = 5Zi<i< n a,-*, for some scalars a,-. Then,
* r X* = Z!i<,-< n o$x?Xxi + YJ\<i<j<na>ai(xJXxi + *J**i) = 0' This implies that
X = 0; indeed, if x is an eigenvector of X for the eigenvalue A., then 0 = xTXx = ^\\x\\2,
yielding A. = 0.
D
The following result of [3] plays a crucial role in the proof, as it will enable us to
characterize the Delaunay transformations of P in Theorem 2.6.
Theorem 2.4 [3] Let P be an n-dimensional Delaunay polytope in Rn with set of vertices
V and such that 0 e V. Let FP denote the smallest face of the cone H\v\ containing dp.
(i) Let A 6 GL(n). If A(P) is a Delaunay polytope, then the vector d A ( P ) e R ( 2 J ) defined
by
for all u, v 6 V, belongs to the relative interior of FP.
(ii) Let d be a vector that lies in the relative interior of Fp. Then, there exists A 6 GL(n)
such that A(P) is a Delaunay polytope and d coincides with the point d A ( P ) defined by
(2.5).
Theorem 2.6 Let P be an n-dimensional Delaunay polytope in Rn having the origin as
a vertex and let FP denote the smallest face of H\v\ containing dp. Let A e GL(n). Then,
A(P) is a Delaunay polytope if and only if the vector dA(p) defined by (2.5) lies in the
relative interior of Fp.
Proof: Necessity follows from Theorem 2.4 (i). Conversely, suppose that dA(p) lies in the
relative interior of FP. By Theorem 2.4 (ii), there exists B e GL(n) such that B(P) is a
Delaunay polytope and dA(p) = dB(p). Then, (u — v)TATA(u — v) = (u — v)TBTB(u-v)
for all u, v e V. As V has full dimension n, we deduce from Lemma 2.3 that AT A = BT B.
Hence, ( B A - 1 ) T ( B A - 1 ) = I, i.e., BA-1 is an orthogonal matrix. This shows that the
polytope A(P) can be obtained from B(P) by applying an orthogonal transformation.
Therefore, A(P) is a Delaunay polytope.
D
Corollary 2.7 Let P be an n-dimensional Delaunay polytope in Rn with set of vertices
V and such thatO € V. Let A € GL(n). Then, A(P) is a Delaunay polytope if and only if
the following holds:
(where the sets Ap and Bp define FP as in relation (2.2)).
We conclude with an auxiliary result that will be needed in the next section.
DELAUNAY TRANSFORMATIONS OF A DELAUNAY POLYTOPE
43
Lemma 2.8 Let P be an n-dimensional Delaunay polytope in Rn with set of vertices V
and such that 0 e V. Let A1, ...,Ak e GL(n) and, for h = 1 , . . . , k , l e t dh be defined by
dt,(u, v) = ||AhU — A h v|| 2 for u, v e V. The following assertions are equivalent.
(i) d 1 , . . . , dk are linearly independent.
(ii) A[A T 1 ,..., ATkAk are linearly independent.
Proof: (i) => (ii) Suppose that Y^,\<h<kahATh Ah = 0 for some scalars ah's. Then,
(u - v)T(Y,i<h<kothAlAh)(u - v) = 0, i.e., Y,1< h < k <x h d h (u, v) = 0 for all u, v e V.
Hence, £i< A < t cthdh = 0, implying that ah, = 0 for all h.
(ii) => (i) Suppose that £, 5A < t ahdh = 0. Then, (u - u) T (£i</,< t othAThAh)(u - v) = 0
for all u, v 6 V. As V is full dimensional and contains the origin, we deduce from Lemma
2.3 that £I<A<* cthAThAh = 0. Therefore, ah = 0 for all h.
P
2.2.
77ie cone C/>
By the considerations in Section 2.1, we are led to define the set Cp, which consists of the
n x H symmetric positive semidefinite matrices A/ that satisfy:
(Recall the definition of the sets BF, Ap from (2.1) and (2.2).) Hence, Cp is a closed cone,
o
whose relative interior Cp consists of the symmetric positive definite matrices M that satisfy
(a), and satisfy (b) with strict inequalities. As an immediate consequence of Corollary 2.7,
the set To(P) defined in (1.4) can be rewritten as
We can express the dimension of the cone Cp in terms of the rank of P. Namely,
Proposition 2.10
The dimension of the cone CP is equal to the rank of P.
Proof: Let k denote the rank of P and let p denote the dimension of the cone Cp. As the
face FP has dimension k, we can find k linearly independent points d 1 , . . . , dk lying in the
relative interior of Fp. By Theorem 2.4 (ii) and Corollary 2.7, there exist A 1 , . . . , Ak €
GL(n) such that ATHAh 6 Cp and dh(u, v) = \\Ahu - Ahv\\2 for u, v e V, h = 1 , . . . ,k.
By Lemma 2.8, A f / A i , . . . , A^Ak are linearly independent. This shows that k < p. Now,
as the cone Cp has dimension p, we can find p linearly independent points M\,... ,Mpm
the relative interior of Cp. Since MI, is positive definite, it has the form Mh = A[Ah for
some Ah e GL(n),for h = I,..., p. Then, the point dh, defined from (2.5) using Ah lies in
the relative interior of FP, for h = 1 , . . . , p. Moreover, d 1 , . . . , dp are linearly independent
by Lemma 2.8. This shows that p < k. Hence, we have the equality: p = k.
D
44
2.3.
LAURENT
The homeomorphism 0
We show here that Theorem 1.11 holds, i.e., that the dimension of the set T(P)/~ is equal
to the rank of P. Recall from (1.5) that the set T(P)/ ~ coincides with the set T 0 (P)/ ~,
where T0(P) is defined by relations (1.4) or (2.9). Consider the mapping
where Mn denotes the set of n x n matrices and PSDn the set of n x n positive semidefinite
matrices. By definition of the equivalence relation ~ from (1.3), we have a bijection
where A denotes the class of A e Mn in the quotient set ,M n /~.
Lemma 2.11 0 is a homeomorphism between the sets Mn/ ~ and PSDn.
Proof: The mapping 9 is clearly continuous. We show that its inverse 0-1 is also continuous. For this, we show that the image 0(C) of any closed set C in Mn/ ~ is a closed
set. Indeed, consider a sequence (A1 ),SN of matrices of Mn for which the class A' of A' in
M n /~ belongs to C for all i e N, and the sequence ((A')T A')ieN is convergent, with limit
M e PSDn. We show that M e 6(C). As the sequence ((A i ) T A i )/eN is convergent, this
implies easily that all the entries of the matrices A' (i e N) are bounded. Hence, we can find
a convergent subsequence (A'OyeN of (A')isN- Denote by A e Mn the limit of (A'OjeNTherefore, A belongs to C, since A1' e C for all j € N and C is closed. Moreover, the
sequence ((A'-O^A'-OyeN converges to Ar A, from which we deduce that
This shows that M = B(A) belongs to <9(C).
n
o
Corollary 2.12 The spaces T0(P)/ ~ and Cp are homeomorphic, via the mapping 0.
Proof: This follows from Lemma 2.11, as the mapping 9 is one-to-one between the sets
T o (P)/~andCp.
D
o
Therefore, both sets T o (P)/ ~ and Cp have the same dimension, which is equal to the
rank of P, by Proposition 2.10. This concludes the Proof of Theorem 1.11.
We conclude with two remarks. The first one illustrates the difficulties encountered when
trying to compute the usual linear dimension of the quotient set T(P)/ ~. The second one
shows that, although 9 extends to a homeomorphism between the closure of the set T(P)/ ~
and the cone Cp, this yields no further result in terms of Delaunay transformations of P.
Remark 2.13 Quite naturally, one may wonder why we did not try to compute the usual
linear dimension of the set T(P)/ ~ (i.e., its maximum number of linearly independent
45
DELAUNAY TRANSFORMATIONS OF A DELAUNAY POLYTOPE
points). It turns out however that this is not a well defined notion as it depends on the choice
of the representatives in the equivalence classes of T(/ > )/~.
To see it, consider again the case of the square y2 = [0, 1]2 from Example 1.7. The
matrices
are Delaunay transformations
of 72, which belong to distinct classes in the quotient set T(y2)/ ~. Clearly, A3 is an
orthogonal matrix, i.e., belongs to the same class as the identity matrix /. Observe now
that A1 , A2, A3 are linearly independent, while the set {A,, A2, I} has rank 2. This shows
that the rank depends on the representatives we use in each class.
Remark 2.14
The closure cl(T0(P)) of the set T0(P) is defined by
As the mapping 0 is one-to-one between the sets d(To(P))/ ~ and Cp, we deduce that these
two sets are homeomorphic. Therefore, the dimension of the set cl(To(P))/ ~ is also equal
to the rank of P. Note, however, that the set cl(T0(P))/ ~ has no immediate interpretation
in terms of Delaunay transformations of P. In particular, the analogue of (1.4) does not
hold, i.e., it is not true that, for any A € Mn,
For instance, take for P the unit square [0, 1]2, with vertices V0 = (0,0), v1 = (1,0),
v2 = (0,1), v3 = (1, 1), and let A := (| j). Then, Av0 = (0,0), Av1 = Av2 = (1, 1),
Av3 = (2, 2), and A(P) is the segment [(0,0), (2,2)]. Hence, A (P) is a Delaunay poly tope,
but the matrix AT A does not belong to the cone Cp. Indeed, points of Cp should satisfy
the triangle equality: xV1V2 = xv0v1 + xv0v2 (because dp satisfies it, as ||v1 — v2||2 =
II"o - fi II2 + ||v0 - v2||2), but 0 = ||Av1 - Av2||2 = ||Av0 - Av1 ||2 + ||Av0 - Av2||2 = 2.
Conversely, if we choose for P an equilateral triangle and for A a transformation of R2
mapping P on a triangle with a right angle, then A(P) is not a Delaunay polytope, while
the matrix AT A clearly belongs to the cone Cp. (In these examples, we use the fact that, for
|V| < 4, the hypermetric cone H\v\is defined by the triangle inequalities: xuv < xuw + xvw
for distinct M, v, w e V.)
Acknowledgments
I thank V.P. Grishukhin and A. Schrijver for some useful conversations on this topic.
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LAURENT
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